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23
A SUFFICIENT AND NECESSARY CONDITION FOR HALPERNTYPE STRONG CONVERGENCE TO FIXED POINTS OF NONEXPANSIVE MAPPINGS
, 2007
"... In this paper, we prove a Halperntype strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983. ..."
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In this paper, we prove a Halperntype strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983.
On the method of cyclic projections for convex sets in Hilbert space
, 1994
"... The method of cyclic projections is a powerful tool for solving convex feasibility problems in Hilbert space. Although in many applications, in particular in the field of image reconstruction (electron microscopy, computed tomography), the convex constraint sets do not necessarily intersect, the met ..."
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The method of cyclic projections is a powerful tool for solving convex feasibility problems in Hilbert space. Although in many applications, in particular in the field of image reconstruction (electron microscopy, computed tomography), the convex constraint sets do not necessarily intersect, the method of cyclic projections is still employed. Results on the behaviour of the algorithm for this general case are improved, unified, and reviewed. The analysis relies on key concepts from convex analysis and the theory of nonexpansive mappings. The notion of the angle of a tuple of subspaces is introduced. New linear convergence results follow for the case when the constraint sets are closed subspaces whose orthogonal complements have a closed sum; this holds, in particular, for hyperplanes or in Euclidean space. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 6502, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and p...
E.: Convergence of stringaveraging projection schemes for inconsistent convex feasibility problems
 Optim. Methods Softw
, 2003
"... We study iterative projection algorithms for the convex feasibility problem of Þnding a point in the intersection of Þnitely many nonempty, closed and convex subsets in the Euclidean space. We propose (without proof) an algorithmic scheme which generalizes both the stringaveraging algorithm and the ..."
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Cited by 4 (2 self)
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We study iterative projection algorithms for the convex feasibility problem of Þnding a point in the intersection of Þnitely many nonempty, closed and convex subsets in the Euclidean space. We propose (without proof) an algorithmic scheme which generalizes both the stringaveraging algorithm and the blockiterative projections (BIP) method with Þxed blocks and prove convergence of the stringaveraging method in the inconsistent case by translating it into a fully sequential algorithm in the product space.
KAMKE'S UNIQUENESS THEOREM
"... A generalization of Kamke's uniqueness theorem in ordinary differential equations is obtained for the limit Cauchy problem, viz x'{t) = f(t, x(t)), x{t)> x0 as 1J10, where / and x take values in an arbitrary normed linear space X and the initial point {t0, x0) is permitted to be on th ..."
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A generalization of Kamke's uniqueness theorem in ordinary differential equations is obtained for the limit Cauchy problem, viz x'{t) = f(t, x(t)), x{t)> x0 as 1J10, where / and x take values in an arbitrary normed linear space X and the initial point {t0, x0) is permitted to be on the boundary of the domain of/. Kamke's hypothesis that \\f(t,x)f{t,y)\ \ < <(>(\tto\, x,y) is replaced by a weaker dissipativetype hypothesis formulated in terms of the duality map of X and a semiinner product derived from it. Even in the scalar case in which X = U, the generalization obtained is still an extension of Kamke's theorem and some of its later analogues. 1.
The approximate fixed point property in Hausdorff topological vector spaces and applications
, 810
"... Abstract. Let C be a compact convex subset of a Hausdorff topological vector space (E, τ) and σ another Hausdorff vector topology in E. We establish an approximate fixed point result for sequentially continuous maps f: (C,σ) → (C,τ). As application, we obtain the weakapproximate fixed point propert ..."
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Abstract. Let C be a compact convex subset of a Hausdorff topological vector space (E, τ) and σ another Hausdorff vector topology in E. We establish an approximate fixed point result for sequentially continuous maps f: (C,σ) → (C,τ). As application, we obtain the weakapproximate fixed point property for demicontinuous selfmapping weakly compact convex sets in general Banach spaces and use this to prove new results in asymptotic fixed point theory. These results are also applied to study the existence of limitingweak solutions for differential equations in reflexive Banach spaces.
THE SET OF COMMON FIXED POINTS OF A ONEPARAMETER CONTINUOUS SEMIGROUP OF MAPPINGS Is ...
, 2004
"... In this paper, we prove the following theorem: Let {T(t) : t ≥ 0} be a oneparameter continuous semigroup of mappings on a subset C of a Banach space E. The set of fixed points of T(t) is denoted by F ( T(t) ) for each t ≥ 0. Then F ( T(t) ) = F ( T(1) ) ∩ F ( T ( √ 2)) t≥0 holds. Using this the ..."
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In this paper, we prove the following theorem: Let {T(t) : t ≥ 0} be a oneparameter continuous semigroup of mappings on a subset C of a Banach space E. The set of fixed points of T(t) is denoted by F ( T(t) ) for each t ≥ 0. Then F ( T(t) ) = F ( T(1) ) ∩ F ( T ( √ 2)) t≥0 holds. Using this theorem, we discuss convergence theorems to a common fixed point of {T(t) : t ≥ 0}.
Stability and Performances in Biclustering Algorithms
"... Abstract. Stability is an important property of machine learning algorithms. Stability in clustering may be related to clustering quality or ensemble diversity, and therefore used in several ways to achieve a deeper understanding or better confidence in bioinformatic data analysis. In the specific f ..."
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Abstract. Stability is an important property of machine learning algorithms. Stability in clustering may be related to clustering quality or ensemble diversity, and therefore used in several ways to achieve a deeper understanding or better confidence in bioinformatic data analysis. In the specific field of fuzzy biclustering, stability can be analyzed by porting the definition of existing stability indexes to a fuzzy setting, and then adapting them to the biclustering problem. This paper presents work done in this direction, by selecting some representative stability indexes and experimentally verifying and comparing their properties. Experimental results are presented that indicate both a general agreement and some differences among the selected methods. 1
Nonexpansive mappings defined on unbounded domains
 Article ID 82080
"... We obtain fixed point theorems for nonexpansive mappings defined on unbounded sets. Our assumptions are weaker than the asymptotically contractive condition recently introduced by JeanPaul Penot. Copyright © 2006 A. Kaewcharoen and W. A. Kirk. This is an open access article distributed under the Cr ..."
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We obtain fixed point theorems for nonexpansive mappings defined on unbounded sets. Our assumptions are weaker than the asymptotically contractive condition recently introduced by JeanPaul Penot. Copyright © 2006 A. Kaewcharoen and W. A. Kirk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
ITERATIVE ALGORITHM FOR A CONVEX FEASIBILITY PROBLEM
"... The purpose of this paper is to study convex feasibility problems in the setting of a real Hilbert space. The approximation of common elements of solution set of variational inequality problems and fixed point set of nonexpansive mappings is considered. Strong convergence theorems are established in ..."
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The purpose of this paper is to study convex feasibility problems in the setting of a real Hilbert space. The approximation of common elements of solution set of variational inequality problems and fixed point set of nonexpansive mappings is considered. Strong convergence theorems are established in the framework of Hilbert spaces. 1